Vulture family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The vulture family of temperaments tempers out the vulture comma (monzo: [24 -21 4⟩, ratio: 10 485 760 000 / 10 460 353 203), a small 5-limit comma of 4.2 cents that is the amount by which a stack of four syntonic commas falls short of the 256/243 Pythagorean limma. As their defining feature, vulture temperaments split the interval 3/1 into four segments (identified in the 5-limit as 320/243).

Vulture

The generator of the vulture temperament is a grave fourth of 320/243, that is, a perfect fourth minus a syntonic comma. Four of these make a perfect twelfth. Its ploidacot is alpha-tetracot. It is a member of the syntonic–diatonic equivalence continuum with n = 4, so it equates a Pythagorean limma with a stack of four syntonic commas. It is also in the schismic–Mercator equivalence continuum with n = 4, so unless 53edo is used as a tuning, the schisma is always observed.

Subgroup: 2.3.5

Comma list: 10485760000/10460353203

Mapping: [⟨1 0 -6], ⟨0 4 21]]

mapping generators: ~2, ~320/243

Optimal tunings:

• WE: ~2 = 1199.9430 ¢, ~320/243 = 475.5200 ¢

error map: ⟨-0.057 +0.125 -0.051]

• CWE: ~2 = 1200.0000 ¢, ~320/243 = 475.5396 ¢

error map: ⟨0.000 +0.203 +0.018]

Optimal ET sequence: 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b

Badness (Sintel): 0.972

Overview to extensions

Temperaments discussed elsewhere include buzzard. Considered below are septimal vulture, terture, condor, eagle, and turkey.

Septimal vulture

Septimal vulture can be described as the 53 & 270 microtemperament, tempering out the ragisma, 4375/4374 and the garischisma, 33554432/33480783 ([25 -14 0 -1⟩) aside from the vulture comma. 270edo is an excellent tuning for this temperament, with generator 107\270. Other compatible tunings include 217edo and 323edo. The harmonic 7 is found at -14 fifths or (-14) × 4 = -56 generator steps, so the smallest mos scale that includes it is the 58-note one, though for larger scope of harmony, you could try the 111- or 164-note one. For a much simpler mapping of 7 at the cost of higher error, you could try buzzard.

It can be extended to the 11-limit by identifying a stack of four 5/4's as 11/9, tempering out 5632/5625, and to the 13-limit by identifying the hemitwelfth as 26/15, tempering out 676/675. Furthermore, the generator of vulture is very close to 25/19; a stack of three generator steps octave-reduced thus represents its fifth complement, 57/50. This corresponds to tempering out 1216/1215 with the effect of equating the schisma with 513/512 and 361/360 in addition to many 11- and 13-limit commas. 270edo remains an excellent tuning in all cases.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 33554432/33480783

Mapping: [⟨1 0 -6 25], ⟨0 4 21 -56]]

Optimal tunings:

• WE: ~2 = 1199.9050 ¢, ~320/243 = 475.5135 ¢

error map: ⟨-0.095 +0.099 +0.039 +0.044]

• CWE: ~2 = 1200.0000 ¢, ~320/243 = 475.5515 ¢

error map: ⟨0.000 +0.251 +0.267 +0.292]

Optimal ET sequence: 53, 164, 217, 270, 593, 863, 1133, 1996d

Badness (Sintel): 0.936

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 41503/41472

Mapping: [⟨1 0 -6 25 -33], ⟨0 4 21 -56 92]]

Optimal tunings:

• WE: ~2 = 1199.9392 ¢, ~320/243 = 475.5326 ¢
• CWE: ~2 = 1200.0000 ¢, ~320/243 = 475.5655 ¢

Optimal ET sequence: 53, 217, 270, 2107c, 2377bc

Badness (Sintel): 1.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4096/4095, 4375/4374

Mapping: [⟨1 0 -6 25 -33 -7], ⟨0 4 21 -56 92 27]]

Optimal tunings:

• WE: ~2 = 1199.9695 ¢, ~154/117 = 475.5451 ¢
• CWE: ~2 = 1200.0000 ¢, ~154/117 = 475.5571 ¢

Optimal ET sequence: 53, 217, 270

Badness (Sintel): 0.775

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728

Mapping: [⟨1 0 -6 25 -33 -7 -12], ⟨0 4 21 -56 92 27 41]]

Optimal tunings:

• WE: ~2 = 1199.9636 ¢, ~25/19 = 475.5426 ¢
• CWE: ~2 = 1200.0000 ¢, ~25/19 = 475.5569 ¢

Optimal ET sequence: 53, 217, 270

Badness (Sintel): 0.579

Semivulture

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 33554432/33480783

Mapping: [⟨2 0 -12 50 41], ⟨0 4 21 -56 -43]]

mapping generators: ~99/70, ~320/243

Optimal tunings:

• WE: ~99/70 = 599.9594 ¢, ~320/243 = 475.5174 ¢
• CWE: ~99/70 = 600.0000 ¢, ~320/243 = 475.5501 ¢

Optimal ET sequence: 106, 164, 270, 916, 1186, 1456

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 3025/3024, 4096/4095, 4375/4374

Mapping: [⟨2 0 -12 50 41 -14], ⟨0 4 21 -56 -43 27]]

Optimal tunings:

• WE: ~99/70 = 599.9859 ¢, ~320/243 = 475.5423 ¢
• CWE: ~99/70 = 600.0000 ¢, ~320/243 = 475.5536 ¢

Optimal ET sequence: 106, 164, 270

Badness (Sintel): 1.47

Terture

Named by Xenllium in 2021, terture tempers out 250047/250000, the landscape comma, and may be described as the 111 & 159 temperament, with a ploidacot signature of triploid gamma-tetracot.

Subgroup: 2.3.5.7

Comma list: 250047/250000, 359661568/358722675

Mapping: [⟨3 0 -18 -32], ⟨0 4 21 34]]

mapping generators: ~63/50, ~320/243

Optimal tunings:

• WE: ~63/50 = 399.9723 ¢, ~320/243 = 475.5221 ¢ (~392/375 = 75.5499 ¢)

error map: ⟨-0.083 +0.134 +0.151 -0.185]

• CWE: ~63/50 = 400.0000 ¢, ~320/243 = 475.5519 ¢ (~392/375 = 75.5519 ¢)

error map: ⟨0.000 +0.253 +0.276 -0.061]

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 2.21

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 19712/19683, 102487/102400

Mapping: [⟨3 0 -18 -32 8], ⟨0 4 21 34 2]]

Optimal tunings:

• WE: ~63/50 = 399.9902 ¢, ~320/243 = 475.5383 ¢ (~392/375 = 75.5481 ¢)
• CWE: ~63/50 = 400.0000 ¢, ~320/243 = 475.5490 ¢ (~392/375 = 75.5490 ¢)

Optimal ET sequence: 111, 159, 270, 1239, 1509, 1779, 2049, 2319

Badness (Sintel): 0.969

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976

Mapping: [⟨3 0 -18 -32 8 -21], ⟨0 4 21 34 2 27]]

Optimal tunings:

• WE: ~63/50 = 399.9958 ¢, ~154/117 = 475.5485 ¢ (~117/112 = 75.5527 ¢)
• CWE: ~63/50 = 400.0000 ¢, ~154/117 = 475.5531 ¢ (~117/112 = 75.5531 ¢)

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 0.771

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913

Mapping: [⟨3 0 -18 -32 8 -21 -2], ⟨0 4 21 34 2 27 12]]

Optimal tunings:

• WE: ~34/27 = 399.9664 ¢, ~112/85 = 475.5198 ¢ (~117/112 = 75.5534 ¢)
• CWE: ~34/27 = 400.0000 ¢, ~112/85 = 475.5568 ¢ (~117/112 = 75.5568 ¢)

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 0.953

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615

Mapping: [⟨3 0 -18 -32 8 -21 -2 -36], ⟨0 4 21 34 2 27 12 41]]

Optimal tunings:

• WE: ~34/27 = 399.9665 ¢, ~112/85 = 475.5198 ¢ (~95/91 = 75.5533 ¢)
• CWE: ~34/27 = 400.0000 ¢, ~112/85 = 475.5568 ¢ (~95/91 = 75.5568 ¢)

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 0.846

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215

Mapping: [⟨3 0 -18 -32 8 -21 -2 -36 10], ⟨0 4 21 34 2 27 12 41 3]]

Optimal tunings:

• WE: ~34/27 = 400.0026 ¢, ~112/85 = 475.5510 ¢ (~24/23 = 75.5485 ¢)
• CWE: ~34/27 = 400.0000 ¢, ~112/85 = 475.5482 ¢ (~24/23 = 75.5482 ¢)

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 1.07

Condor

Condor tempers out 10976/10935 and may be described as the 58 & 159 temperament. The generator represents the septimal diminished fifth (112/81), and three minus an octave make vulture's generator of ~320/243. The ploidacot for this temperament is epsilon-dodecacot. 217edo is an excellent tuning for this temperament.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 40353607/40000000

Mapping: [⟨1 -4 -27 -20], ⟨0 12 63 49]]

mapping generators: ~2, ~112/81

Optimal tunings:

• WE: ~2 = 1200.0142 ¢, ~112/81 = 558.5276 ¢

error map: ⟨+0.014 +0.319 +0.539 -1.260]

• CWE: ~2 = 1200.0000 ¢, ~112/81 = 558.5212 ¢

error map: ⟨0.000 +0.300 +0.523 -1.287]

Optimal ET sequence: 58, 159, 217

Badness (Sintel): 3.92

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 10976/10935

Mapping: [⟨1 -4 -27 -20 -24], ⟨0 12 63 49 59]]

Optimal tunings:

• WE: ~2 = 1199.9730 ¢, ~112/81 = 558.5052 ¢
• CWE: ~2 = 1200.0000 ¢, ~112/81 = 558.5173 ¢

Optimal ET sequence: 58, 101cd, 159, 217, 376d

Badness (Sintel): 1.60

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 10976/10935

Mapping: [⟨1 -4 -27 -20 -24 -34], ⟨0 12 63 49 59 81]]

Optimal tunings:

• WE: ~2 = 1199.9649 ¢, ~112/81 = 558.5040 ¢
• CWE: ~2 = 1200.0000 ¢, ~112/81 = 558.5197 ¢

Optimal ET sequence: 58, 159, 217

Badness (Sintel): 1.05

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 676/675, 8624/8619

Mapping: [⟨1 -4 -27 -20 -24 -34 12], ⟨0 12 63 49 59 81 -17]]

Optimal tunings:

• WE: ~2 = 1199.9594 ¢, ~112/81 = 558.5017 ¢
• CWE: ~2 = 1200.0000 ¢, ~112/81 = 558.5202 ¢

Optimal ET sequence: 58, 159, 217

Badness (Sintel): 1.12

Eagle

Eagle tempers out 2401/2400 and may be described as the 58 & 270 temperament. It has a semi-octave period and a generator of ~28/27, four of which make a hemifourth which may be identified with 15/13, and two of those make a perfect fourth; its ploidacot thus is diploid wau-octacot. Compatible tunings include 212edo, 270edo, and 328edo.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 10485760000/10460353203

Mapping: [⟨2 4 9 8], ⟨0 -8 -42 -23]]

mapping generators: ~177147/125440, ~28/27

Optimal tunings:

• WE: ~177147/125440 = 599.9818 ¢, ~28/27 = 62.2266 ¢

error map: ⟨-0.036 +0.159 +0.004 -0.184]

• CWE: ~177147/125440 = 600.0000 ¢, ~28/27 = 62.2295 ¢

error map: ⟨0.000 +0.209 +0.046 -0.105]

Optimal ET sequence: 58, 154c, 212, 270, 752, 1022, 1292, 2854b

Badness (Sintel): 1.51

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 19712/19683

Mapping: [⟨2 4 9 8 12], ⟨0 -8 -42 -23 -49]]

Optimal tunings:

• WE: ~99/70 = 599.9796 ¢ ~28/27 = 62.2218 ¢
• CWE: ~99/70 = 600.0000 ¢, ~28/27 = 62.2251 ¢

Optimal ET sequence: 58, 154ce, 212, 270

Badness (Sintel): 0.823

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 10648/10647

Mapping: [⟨2 4 9 8 12 13], ⟨0 -8 -42 -23 -49 -54]]

Optimal tunings:

• WE: ~99/70 = 599.9763 ¢ ~28/27 = 62.2174 ¢
• CWE: ~99/70 = 600.0000 ¢, ~28/27 = 62.2211 ¢

Optimal ET sequence: 58, 154cef, 212, 270

Badness (Sintel): 0.673

Turkey

Named by Xenllium in 2021, turkey may be described as the 212 & 217 temperament. It is generated by a fifth sharp of just, close to 3\5 but on the flat side thereof, which can be interpreted as 50/33 in the 11-limit. Sixteen generators minus nine octaves make a perfect fifth; its ploidacot is thus theta-16-cot. 429edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 5250987/5242880

Mapping: [⟨1 -8 -48 7], ⟨0 16 84 -7]]

mapping generators: ~2, ~3592/1715

Optimal tunings:

• WE: ~2 = 1200.1147 ¢, ~3592/1715 = 718.9483 ¢

error map: ⟨+0.115 +0.300 -0.161 -0.661]

• CWE: ~2 = 1200.0000 ¢, ~3592/1715 = 718.8806 ¢

error map: ⟨0.000 +0.134 -0.345 -0.990]

Optimal ET sequence: 212, 429, 1070d

Badness (Sintel): 5.34

11-limit

Subgroup: 2.3.5.7.11

Comma list: 19712/19683, 42875/42768, 160083/160000

Mapping: [⟨1 -8 -48 7 -87], ⟨0 16 84 -7 151]]

Optimal tunings:

• WE: ~2 = 1200.1131 ¢ ~50/33 = 718.9478 ¢
• CWE: ~2 = 1200.0000 ¢, ~50/33 = 718.8808 ¢

Optimal ET sequence: 212, 429

Badness (Sintel): 2.63

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104

Mapping: [⟨1 -8 -48 7 -87 -61], ⟨0 16 84 -7 151 108]]

Optimal tunings:

• WE: ~2 = 1200.1324 ¢ ~50/33 = 718.9608 ¢
• CWE: ~2 = 1200.0000 ¢, ~50/33 = 718.8825 ¢

Optimal ET sequence: 212, 217, 429

Badness (Sintel): 1.81